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1
?1/4
exp ? (?a)(?b) tan?1 (az/bz) ?
?(x) = (az)2 + (bz)2
2
1 ?1
? exp ? i??? 1 + ?2 (?b + ??a) ?
?
2

? ln (az)2 + (bz)2 + 2? tan?1 (az/bz) ?,

where zµ = xµ + ?µ and aµ , bµ , ?µ , ? are arbitrary constants satisfying conditions
(5.16).
(iv) k = 1 ,
2

1 1
? ?1 ?
(?a + ??b)(az + ?bz) + ?c cz + (bz)2
?(x) = exp (?c)(?b)bz
4 4
1
?1
? exp ?i??? ?1 + ?2 ?1 (?a + ??b) + ?2 ?c ?
2 2
?
2
1
? ?1 (az + ?bz) + ?2 cz + (bz)2 ? ?1 ?,
2
1 2
? = (az + ?bz)2 + cz + (bz)2 , zµ = xµ + ?µ ,
4
and ?µ , aµ , bµ , cµ , ?, ?i are arbitrary constants satisfying the conditions

aa = ?1, cc = ?4. (5.17)
ab = bc = ca = bb = 0,

(v) k < 0,

1
?(x) = exp (?c)(?b)bz (?a + ??b)(az + ?bz)+
4
1
+ (?c) cz + (bz)2 f (?) + ig(?) ?,
4
1 2
? = (az + ?bz)2 + cz + (bz)2
zµ = xµ + ?µ ,
4
with f (?), g(?) from (5.8). Parameters aµ , bµ , cµ , ?µ satisfy conditions (5.17) and ?
is an arbitrary constant spinor.
(vi) k ? R1 , k = 0,

1 1
?(x) = exp (?a)(?b) ln(az + bz) exp (?c)(?a + ?b) +
2 2
(5.18)
+ i?(??)1/2k (?c ? ?a ? ?b) [ln(az + bz) ? cz] ?,
?
278 W.I. Fushchych, R.Z. Zhdanov

1
(?a)(?b) ln(az + bz) ?
?(x) = exp
2
(5.19)
1
? exp (?c)(?a + ?b) ? i?(??)1/2k ?c (cz) ?,
?
2
where zµ = xµ + ?µ , ? is an arbitrary constant spinor and aµ , bµ , cµ are arbitrary
constants satisfying conditions
?aa = bb = ?1, cc = ?1, ab = bc = ca = 0.
(vii) k ? R1 , k = 0,
1
(?a)(?b)bz exp ?i?(?c + ??b)(??)1/2k (cz + ?bz) ?, (5.20)
?(x) = exp ?
2
where zµ = xµ + ?µ , ? is an arbitrary constant spinor and aµ , bµ , cµ , ?µ are arbitrary
constants satisfying the conditions
aa = cc = ?1, (5.21)
ab = bc = ca = bb = 0.
In conclusion of this section, let us consider the special case of equation (1.1) when
k = 3 . It is common knowledge that the corresponding non-linear Dirac equation is
2
conformally invariant [10, 13]. This enables us to obtain a larger family of solutions
with the help of a procedure of generating solutions by special conformal transforma-
tions, corresponding formulae having the form [7]
?2 (x) = ? ?2 (x)(1 ? (?x)(??))?1 (x ),
(5.22)
xµ = (xµ ? ?µ (xx))? ?1 (x), ?(x) = 1 ? 2?x + (??)(xx).
3
Using solutions (5.14) under k = as ?1 (x) we obtain a new solution of the
2
conformally invariant equation (1.1)
1?
?(x) = [1 ? (?x)(??)]? ?2 (x) exp ?(?a)(?b)(bx ? (b?)(xx))? ?1 (x) ?
2
1
? exp ? i?(??)1/3 ?a[2(ax ? (a?)(xx))?(x) +
? (5.23)
2

+ ?(bx ? (b?)(xx))2 ]? ?2 (x) ?,
?

?
where aa = ?1, bb = 0, ab = 0 and ?µ , ? are arbitrary constants.
3
The same procedure when applied to solutions (5.18)–(5.20) under k = give
2
some new solutions of the non-linear Dirac equation.

6. Exact solutions of the system (1.2)
We shall seek solutions of (1.2) when m1 = 0, m2 = 0, the following ansatz being
used:
?(x) = ?b exp(if (ax))?,
(6.1)
Aµ (x) = bµ g1 (ax) + aµ g2 (ax),
where bb = 0, ax = aµ xµ and f , g1 , g2 are arbitrary differentiable functions.
On some exact solutions of a system of non-linear differential equations 279

Substitution of (6.1) into (1.2) gives the system of ODE

?1 g2 = f?,
(aa)?1 = ?2eb? ? ?2 g1 2abg1 g2 + (aa)g2 ,
2 (6.2)
g
?(ab)?1 = ??2 g2 2abg1 g2 + (aa)g2 ,
2
g

where a dot means differentiation with respect to ? = ax, b? = bµ ?µ , aa = aµ aµ ,
ab = 0, ?µ = ??µ ?, µ = 0, 3.
?
We have succeeded in integrating the system (6.2) in the case aa = 0, ab = 0, i.e.

?1 g2 = f?,
g2 g1 = ?(eb?)/(?2 ab),
2 (6.3)
2
g1 = 2?2 g1 g2 .
?

From the second equation it follows that
?2
g2 = ?(eb?)/(?2 ab)g1 . (6.4)

Substituting (6.4) into (6.3) we obtain ODE for determination of g1 (?)
v
?3
g1 = k 2 /?2 g1 , k = 2(eb?)/(ab). (6.5)
?

Integration of the last ODE yields

2|?2 |1/2 |k|?1 g1 ,
2
?2 < 0,
(6.6)
? + C2 = 1/2
?1
C1 C1 g1 ? k 2 /?2
2
, C1 = 0.

Finally
?1
?1/2
g1 = ±C1 (C1 ? + C2 )2 + k 2 /?2 (6.7)
C1 = 0, ,
?1
g1 = ±(k/|?2 |) 2|k||?2 |?1/2 ? + C2 (6.8)
?2 < 0, .

Substituting the above results into (6.4) we find expressions for g2 (?)
?1
g2 = ?(kC1 /?2 ) (C1 ? + C2 )2 + k 2 /?2 (6.9)
C1 = 0, ,
?1
g2 = ?(k/|?2 |) 2|k||?2 |?1/2 ? + C2 (6.10)
?2 < 0, .

Substituting these expressions into the first equation from (6.3) we obtain f (?)
?1/2
tan?1 k ?1 ?2 (C1 ? + C2 ) ,
1/2
f (?) = ??1 ?2 (6.11)
C1 = 0,

f (?) = ?1 |?2 |?1/2 ln 2k|?2 |?1/2 ? + C2 , (6.12)
?2 < 0,

where C1 , C2 are arbitrary constants.
Substitution of (6.7)–(6.12) into (6.1) gives two families of solutions of the initial
equation (1.2)
280 W.I. Fushchych, R.Z. Zhdanov

(i) ?2 = 0, C1 = 0,
?1/2
tan?1 ?2 k ?1 (C1 ax + C2 )
1/2
?(x) = ?b exp ?i?1 ?2 ?,
?1/2 1/2
(C1 ax + C2 )2 ? k 2 ??1 (6.13)
Aµ (x) = ±bµ C1 ?
2
?1
? aµ (kC1 /?2 ) (C1 ax + C2 )2 ? k 2 /?2 ,
(ii) ?2 < 0,

?(x) = ?b exp ?i?1 |?2 |?1/2 ln 2k|?2 |?1/2 ax + C3 ?,
(6.14)
?1
1/2
?1/2 ?1/2
Aµ (x) = ±bµ 2k|?2 | ? aµ (k/|?2 |) 2k|?2 |
ax + C3 ax + C3 ,
v
where k = 2ebµ (??µ ?)/(ab), C1 , C2 , C3 are arbitrary constants and ? is an arbitrary
?
constant spinor.
Let us note that the solutions obtained depend analytically on parameters ?1 , e
while parameter ?2 is included in a singular way. It means that solutions (6.13) and
(6.14) cannot be obtained in the framework of perturbation theory by expanding in a
series with respect to a small parameter ?2 .
On introducing as usual the tensor of the electromagnetic field Fµ? = ?A? /?xµ ?
?Aµ /?x? we obtain
?1/2
1/2
Fµ? = ±(aµ b? ? a? bµ )C1 (C1 ax + C2 )2 ? k 2 /?2 ,
?1/2
Fµ? = ±(aµ b? ? a? bµ )k|?2 |?1/2 2k|?2 |?1/2 ax + C3
for solutions (6.13) and (6.14) respectively.
To obtain new families of solutions of the system (1.2) one can use its symmetry
under conformal group C(1, 3) [9]. The formula for generating solutions by special
conformal transformations has the form [8]
?2 (x) = ? ?2 [1 ? (?x)(??)]?1 (x ),
Aµ (x) = ? ?2 (x)[gµ? ?(x) + 2(?µ x? ? ?? xµ + 2?xxµ ?? ?
(2)

? xx?µ ?? ? ??xµ x? )]A? (x ),
(1)
xµ = (xµ ? ?µ xx)? ?1 (x), ?(x) = 1 ? 2?x + (??)(xx).
(1)
Using (6.13) and (6.14) as ?1 (x) and Aµ (x) one can construct new multiparame-
ter families of exact solutions of (1.2) but we omit corresponding formulae because of
their cumbersome character.

7. Conclusion
In the present work, large classes of exact solutions of the non-linear Dirac
equation and of the system of non-linear equations of quantum electrodynamics
were constructed. Solutions obtained by Akdeniz [1], Fushchych and Shtelen [6, 7],
Kortel [11], Merwe [14] and Takahashi [15] can be obtained with the help of ans?tzea
(3.16)–(3.29).
Most of the solutions depend analytically on constants ?, ?i , e. However solutions
(6.13) and (6.14) have a non-perturbative character because of their singular depen-
dence on the parameter ?2 .
On some exact solutions of a system of non-linear differential equations 281

We have constructed ans?tze which reduced the four-dimensional systems (1.1)
a
and (1.2) to three-, two- and one-dimensional systems of PDE. It is important to note
that these ansa? can be applied to any spinor equations which are invariant under
tze
the extended Poincar? group P(1, 3).
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